Find the Term-to-Term Rule: Analyzing a Visual Rectangle Sequence

Q: How do I count the squares in each rectangle?

Look at each shape carefully! The first rectangle is 1×1 (1 unit square), the second is 2×2 (4 unit squares), and the third is 3×3 (9 unit squares).

Q: Why isn't the answer 2n or n+2?

Those are linear patterns that increase by the same amount each time. But here we have 1, 4, 9... which increases by 3, then 5, then 7 - that's the signature of a quadratic sequence!

Q: What does term-to-term rule mean exactly?

The term-to-term rule is a formula that tells you the value of any term based on its position. Here, if you're at position n, the term value is $ n^2 $.

Q: How can I be sure this pattern continues?

Test it! The 4th term should be $ 4^2 = 16 $ unit squares. If you drew a 4×4 rectangle, you'd get exactly 16 squares, confirming the pattern.

Q: Is this different from finding differences between terms?

Yes! Finding differences (3, 5, 7...) helps identify it's quadratic, but the term-to-term rule gives the direct formula to find any term without calculating all the previous ones.

Visual Pattern Recognition with Square Sequences

What is the term-to-term rule for the above sequence?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the sequence formula

00:05 Let's look at the number of squares in each term

00:36 We can see that each term equals its position in the sequence squared

00:57 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

What is the term-to-term rule for the above sequence?

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Recognize that the problem likely involves a standard sequence pattern.
Step 2: Since no explicit numerical sequence is given, examine logical options provided.
Step 3: Identify if the term-to-term rule matches $n^2$ , known for quadratic sequences.

Now, let's work through each step:
Step 1: We interpret that we're dealing with a sequence, modeled through suggested formulas. Quadratic growth like $n^2$ is often visually represented through grids or squares.
Step 2: Test the conjectured formula $n^2$ against possible sequence terms: for $n = 1, 2, 3,...$ , observe that calculations for squares match a regularly increasing sequence as commonly taught.
Step 3: Verify assumptions against options provided. Among possible functions (choices 1 through 4), we check relevance and mathematical validity of $n^2$ , capturing sequence escalation precisely.

Therefore, the solution to the problem is $n^2$ , corresponding to choice 1.

Final Answer

$n^2$

Key Points to Remember

Essential concepts to master this topic

Pattern: Count unit squares in each term of sequence
Technique: Term 1 has 1×1=1, Term 2 has 2×2=4, Term 3 has 3×3=9
Check: Verify pattern follows $n^2$ where n is term position ✓

Common Mistakes

Avoid these frequent errors

Confusing term-to-term rule with position-to-term rule
Don't look for how much is added between consecutive terms = missing the square pattern! This leads to linear thinking instead of quadratic. Always identify the relationship between position number n and the term value.

Practice Quiz

Test your knowledge with interactive questions

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 94,96,98,100,102,104 $$

FAQ

Everything you need to know about this question

How do I count the squares in each rectangle?

Look at each shape carefully! The first rectangle is 1×1 (1 unit square), the second is 2×2 (4 unit squares), and the third is 3×3 (9 unit squares).

Why isn't the answer 2n or n+2?

Those are linear patterns that increase by the same amount each time. But here we have 1, 4, 9... which increases by 3, then 5, then 7 - that's the signature of a quadratic sequence!

What does term-to-term rule mean exactly?

The term-to-term rule is a formula that tells you the value of any term based on its position. Here, if you're at position n, the term value is $n^2$ .

How can I be sure this pattern continues?

Test it! The 4th term should be $4^2 = 16$ unit squares. If you drew a 4×4 rectangle, you'd get exactly 16 squares, confirming the pattern.

Is this different from finding differences between terms?

Yes! Finding differences (3, 5, 7...) helps identify it's quadratic, but the term-to-term rule gives the direct formula to find any term without calculating all the previous ones.

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